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6 Two-Dimensional Problems in Elasticity …
(a) Cylinder under axisymmetric loading
(b) Circular footing on soil mass
Fig. 6.4 Axisymmetric problems
6.8 Thick-Walled Cylinder Subjected to Internal
and External Pressures
Consider a cylinder of inner radius “a” and outer radius “b” as shown in Fig. 6.5. Let
the cylinder be subjected to internal pressure p i and an external pressure p 0 . This
problem can be treated either as a plane stress case (σ z = 0) or as a plane strain case
(ε z = 0).
Case (a): Plane stress.
Consider the ends of the cylinder which are free to expand. As the axial loading is
absent, σ z = 0. The stresses are symmetrical about the z-axis and therefore, τ rθ = 0.
Neglecting the body forces, Eq. (2.46) reduces to
∂σ r
∂r
+
σ r − σ θ
r
= 0
(6.20)